Diffusion and multiplication in random media

We investigate the evolution of a population of non-interacting particles which undergo diffusion and multiplication. Diffusion is assumed to be homogeneous, while multiplication proceeds with different rates reflecting the distribution of nutrients. The distribution of nutrients is considered as a stationary quenched random variable with zero average, so the population size would remain constant if there were no fluctuations in the distribution of nutrients. We show that fluctuations drastically affect the behavior, e.g.?the population size exhibits a super-exponential growth whenever the nutrient distribution is unbounded. We elucidate a huge difference between the average and typical asymptotic growths and emphasize the role played by the spatial correlations in the nutrient distribution.

[1]  L. Ballentine,et al.  Qualitative Methods in Quantum Theory , 1977 .

[2]  Massimo Vergassola,et al.  Bacterial strategies for chemotaxis response , 2010, Proceedings of the National Academy of Sciences.

[3]  E. Gross.  Partition function of a particle subject to Gaussian noise , 1983 .

[4]  Tao.  Exact solution for diffusion in a random potential. , 1988, Physical review letters.

[5]  Havlin,et al.  Comment on "Delocalization in the 1D anderson model with long-range correlated disorder" , 2000, Physical review letters.

[6]  Salinas,et al.  Nonlinear measures for characterizing rough surface morphologies , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  J. Luck,et al.  Lifshitz tails and long-time decay in random systems with arbitrary disorder , 1988 .

[8]  Yicheng Zhang,et al.  Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics , 1995 .

[9]  Ya. B. Zel'Dovich,et al.  Intermittency in random media , 1987 .

[10]  Michael M. Desai,et al.  A quasispecies on a moving oasis. , 2003, Theoretical population biology.

[11]  C. DeWitt-Morette,et al.  Techniques and Applications of Path Integration , 1981 .

[12]  W. Gabriel,et al.  Biological evolution through mutation, selection, and drift: An introductory review , 1999, cond-mat/9907372.

[13]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[14]  H. Berg,et al.  Dynamics of formation of symmetrical patterns by chemotactic bacteria , 1995, Nature.

[15]  Guyer,et al.  Comment on "Exact solution for diffusion in a random potential" , 1990, Physical review letters.

[16]  Sidney Redner,et al.  Random multiplicative processes: An elementary tutorial , 1990 .

[17]  On the statistics of superlocalized states in self-affine disordered potentials , 2004, cond-mat/0409117.

[18]  H. Berg,et al.  Complex patterns formed by motile cells of Escherichia coli , 1991, Nature.

[19]  F. Moura,et al.  Delocalization in the 1D Anderson Model with Long-Range Correlated Disorder , 1998 .

[20]  L. Pastur,et al.  Introduction to the Theory of Disordered Systems , 1988 .

[21]  A. Mikhailov.  Selected topics in fluctuational kinetics of reactions , 1989 .

[22]  S. Redner,et al.  Random walk in a random multiplicative environment , 1989 .

[23]  M. Cates,et al.  Statistics of a polymer in a random potential, with implications for a nonlinear interfacial growth model , 1988 .

[24]  M. Eigen,et al.  The Hypercycle: A principle of natural self-organization , 2009 .

[25]  S. Solomon,et al.  Adaptation of autocatalytic fluctuations to diffusive noise. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  D. Sokoloff,et al.  Self-excitation of a nonlinear scalar field in a random medium. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[27]  D. Nelson,et al.  Life and death near a windy oasis , 1998, Journal of mathematical biology.

[28]  J. Krug,et al.  Directed polymers in the presence of colununar disorder , 1993 .

[29]  M. Eigen.  Selforganization of matter and the evolution of biological macromolecules , 1971, Naturwissenschaften.

[30]  P. Gennes.  Chemotaxis: the role of internal delays , 2004, European Biophysics Journal.

[31]  Michael V Berry,et al.  Nonspreading wave packets , 1979 .

[32]  Werner Ebeling,et al.  Diffusion and reaction in random media and models of evolution processes , 1984 .

[33]  J. K. Anlauf,et al.  Asymptotically Exact Solution of the One-Dimensional Trapping Problem , 1984 .

[34]  P. Zweifel.  Advanced Mathematical Methods for Scientists and Engineers , 1980 .

[35]  W. Ewens.  Mathematical Population Genetics , 1980 .

[36]  M. Isichenko.  Percolation, statistical topography, and transport in random media , 1992 .

[37]  S. Edwards,et al.  The size of a polymer in random media , 1988 .

[38]  REVIEWS OF TOPICAL PROBLEMS: Critical phenomena in media with breeding, decay, and diffusion , 1984 .

[39]  J. Machta,et al.  Polymers in a disordered environment , 1989 .

[40]  S. Redner,et al.  Unimolecular reaction kinetics , 1984 .

[41]  Renz,et al.  Diffusion in a random catalytic environment, polymers in random media, and stochastically growing interfaces. , 1989, Physical review. A, General physics.

[42]  P. Grassberger,et al.  The long time properties of diffusion in a medium with static traps , 1982 .

[43]  M. Matsushita,et al.  Experimental Investigation on the Validity of Population Dynamics Approach to Bacterial Colony Formation , 1994 .

[44]  Engel,et al.  Comment on "Diffusion in a random potential: Hopping as a dynamical consequence of localization" , 1987, Physical review letters.

[45]  H. Leschke,et al.  Long-time asymptotics of diffusion in random media and related problems , 1989 .

[46]  D. Huse,et al.  Pinning and roughening of domain walls in Ising systems due to random impurities. , 1985, Physical review letters.

[47]  Nieuwenhuizen.  Trapping and Lifshitz tails in random media, self-attracting polymers, and the number of distinct sites visited: A renormalized instanton approach in three dimensions. , 1989, Physical review letters.

[48]  O. Schramm,et al.  Contour lines of the two-dimensional discrete Gaussian free field , 2006, math/0605337.

[49]  David R. Nelson,et al.  NON-HERMITIAN LOCALIZATION AND POPULATION BIOLOGY , 1997, cond-mat/9708071.

[50]  J. Krug,et al.  Adaptation in simple and complex fitness landscapes , 2005, q-bio/0508008.

[51]  Zhang.  Diffusion in a random potential: Hopping as a dynamical consequence of localization. , 1986, Physical review letters.

[52]  S. Redner,et al.  A Kinetic View of Statistical Physics , 2010 .